Environmental Engineering Reference
In-Depth Information
where
Q
is the solar constant,
a
is the amplitude of the periodic forcing,
ω
is its
10
−
4
and 2
angular frequency (with
a
(
T
) is the albedo.
Invoking the results of earlier studies, the authors assume that, in the absence of the
periodic forcing (i.e.,
a
=
5
×
π/ω
=
92 kyr), and
α
=
0), dynamics (
4.71
) are bistable, with an unstable state
T
T
1
,
3
. To this end,
they replaced the right-hand side of (
4.70
) [with
R
in
and
R
out
expressed by (
4.71
) with
a
=
T
2
symmetrically located between the two stable states
T
=
=
0] with a function proportional to a third-order polynomial with roots
T
1
,
T
2
,and
T
3
, i.e.,
1
1
1
T
T
1
T
T
2
T
T
3
Q
(1
−
α
(
T
))
−
E
(
T
)
=
E
(
T
)
β
−
−
−
.
(4.72)
Thus, for the generic case
a
=
0, Eq. (
4.70
) becomes
c
d
T
d
t
=
F
(
T
)
=
E
(
T
)
(4.73)
1
1
1
[
a
cos(
t
)
T
T
1
T
T
2
T
T
3
×
β
−
−
−
ω
t
)
+
1]
+
a
cos(
ω
.
The dynamical properties of this system strongly depend on the bistability assump-
tion expressed by (
4.72
). Although
Benzi et al.
(
1982b
) do not explain the physical
rationale underlying this assumption, other studies conjectured that the climate sys-
tem might be bistable (e.g.,
Mason
,
1976
). In the absence of noise, periodic forcing
is unable to induce transitions across the potential barrier for realistic values of the
system's parameters (
T
1
=
4000). The pres-
ence of additive noise may profoundly affect the dynamical properties of the system.
Benzietal.
(
1982b
) considered the effect of environmental variability on dynamics
(
4.73
). In particular, they concentrated on the case of additive white Gaussian noise
ξ
gn
with mean zero and intensity
s
gn
:
d
T
d
t
=
278
.
6K,
T
2
=
283
.
3K,
T
3
=
288 K,
c
=
F
(
T
)
c
+
ξ
.
(4.74)
gn
Notice how this system differs from the one presented in Chapter 3. In fact in this
case the periodic forcing appears as a multiplicative term in the bistable potential.
In the absence of periodic fluctuations (i.e.,
a
0) noise induces transitions be-
tween the wells of the potential. The mean exit time from the domain of attraction of
T
3
can be calculated with
Kramer's method
(see Chapter 3):
=
π
e
V
2
,
3
t
c
2
,
3
≈
√
|
(4.75)
s
gn
V
(
T
2
)
V
(
T
3
)
|
where
V
(
T
) is the potential function defined by (
4.70
)and
V
2
,
3
is the height of the
=−
T
3
potential barrier from the stable state,
T
3
,
V
2
,
3
T
2
F
(
T
)d
T
.
=
0 the periodic forcing modulates the shape of the potential function
and the height of the potential barrier fluctuates between a minimum
When
a
V
min
2
and a
,
3
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